Questions tagged [matching]
A matching (aka **Independent Edge Set**) in a simple graph is the set of pairwise non-adjacent edges i.e. no two edges have common vertex.
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questions
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Matching of two weighted graphs allowing one-to-many mapping
I am looking for a heuristic for a graph matching problem as follows.
Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
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1answer
24 views
Prove that any tree contains a matching of size |InternalNodes|/2 [closed]
How can i prove that any tree contains a matching of size |InternalNodes|/2?
Thanks in advance
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14 views
Matching algorithm to find pattern match in rectangles
I am trying to write an algorithm to find the best (if any) match for a pattern of rectangles. Essentially, given a database of three different arrays of rectangles. The color is irrelevant here, it's ...
4
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1answer
71 views
Term for a graph decomposition based on a maximum matching
Let $M$ be a maximum cardinality matching in a bipartite graph $G(X+Y,E)$. Let $X_0$ be the subset of $X$ unmatched by $M$. Define the following sequence:
$Y_1 = $ the neighbors of $X_0$ using edges ...
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0answers
59 views
Maximum weighted matching for directed (non-bipartite) graphs
This post concerns mainly non-bipartite graphs.
Edmonds (1961) have proposed the Blossom algorithm to solve the maximum matching problem for undirected graphs. The best implementation of it is due ...
4
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1answer
130 views
Matching Algorithm - How to maximize matched quantity with unique matching rules?
Given a set $S=\{A,B,\cdots,H\}$. Elements in $S$ can be matched according to the following rules:
$$\begin{aligned}
A\leftrightarrow B\\
C\leftrightarrow D\\
B+C\leftrightarrow F\\
D+A\...
4
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1answer
31 views
How to construct an ordinary matching from a fractional matching?
Given a graph $G=(V,E)$. A fractional matching, say $f$, assigns every edge $e \in E$ to a fraction $f(e) \in [0,1]$, with the constraint: for $v \in V$, $\sum_{e \ni v}f(e) \leq 1 $.
My question ...
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35 views
Online Many-to-one Matching
Offline Problem
I have a graph $\mathcal{G} = (\mathcal{D} \cup \mathcal{A},
\mathcal{E})$. Each edge $e \in \mathcal{E}$ between the two vertex
sets $\mathcal{D}$ and $\mathcal{A}$ has an ...
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0answers
14 views
Online bipartite matching problem for task assignment
I have $n$ drivers, each one has a balance (in Us dollars), availability status (true if he is not working already) and number of accomplished tasks in the current ...
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0answers
51 views
Perfect matching in complete, weighted graph
I'm trying to find pairs in a complete, weighted graph, similar to the one below (weights not shown).
For each possible pair there is a weight and I would like to find pairs for including all ...
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0answers
21 views
Optimal matching of individuals in vehicles
I am looking for an algorithm to find the optimal matching/allocation of n individuals in m identical vehicles. The aim is to create groups of individuals who will share these vehicles. Groups' size ...
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23 views
Bipartite vertex cover [duplicate]
If this link can be any help
https://codeforces.com/blog/entry/63164
A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set.
A ...
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69 views
Find a minimum weight matching of a specific size
Given a positive-weighted complete graph $G=(V,E)$ and an even integer $k$, I want to find a minimum weight matching of size exactly $k$, i.e., I want to choose $k/2$ vertex disjoint edges such that ...
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3answers
877 views
What is a fractional matching?
For the maximum matching problem, we can find the fractional matching which I understand involves some sort of weighting for the edges. However, I cannot seem to find an exact and simple explanation ...
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88 views
Maximum matching blossom algorithm
https://stanford.edu/~rezab/classes/cme323/S16/projects_reports/shoemaker_vare.pdf
Why do we use blossom contrast in maximum matching?
The examples that I have seen can easily be solved by just ...
3
votes
2answers
42 views
Minimizing catastrophic risk in Gale-Shapley matching
In the hospital-resident assignment problem we have to match a large set of med students with a small set of hospitals. Hospitals may accept multiple students, but the number of students is much ...
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0answers
31 views
Find maximal matching in tree in $O\left(\frac{n}{\log n}\right)$
As any tree can be described as a binary sequence ($i$-th bit is 0 if the edge goes down and 1 otherwise, every edge is travelled twice $-$ up and down, so such sequence's length is $2 |V| - 2$), any ...
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0answers
36 views
Computing a shortest $M$-alternating walk (for the Blossom algorithm)
When explaining the Blossom algorithm for maximum (nonbipartite) matchings, Shrijver describes, given a simple graph $G = (V,E)$, a matching $M \subseteq E$, and the set $X \subseteq V$ of nodes ...
3
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1answer
36 views
Can maximum matching algorithms be used for maximum weight matching?
There are two fast algorithms for maximum matching on general graphs:
Micali and Vazirani in $O(E\sqrt{V})$.
Mucha and Sankowski in $O(V^{2.376})$.
Can these be also used for maximum weighted ...
2
votes
1answer
93 views
Find a minimum-weight perfect b-matching, where b is even
How would one find a minimum-weight perfect b-matching of a general graph, where the number of edges incident on each vertex is a positive even number not greater than b?
A minimum-weight perfect b-...
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1answer
72 views
Find a minimum-weight perfect 2-matching
How would one find a minimum-weight perfect 2-matching of a general graph? Is it possible to use standard matching techniques like Blossom V?
A minimum-weight perfect 2-matching of a graph G is a ...
4
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1answer
382 views
Maximum matching using linear programming
In a bipartite graph $G = (V,E)$, there is a neat algorithm for finding a maximum matching (or even a maximum-weight matching) using linear programming. It is explained here. The first step is to ...
4
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2answers
112 views
Run-time of Hungarian algorithm - matrix formulation
There are many different explanations of the Hungarian algorithm. My favorite explanation is the one based on matrices, for example here, since it is very intuitive and easy to carry out in a ...
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1answer
42 views
Minimal edge cover of the hypergraph
We know that minimal edge cover for the normal graph is polynomial time solvable.
Is it also true for hypergraph?
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1answer
80 views
minimum cardinality maximal matching of graph
How to find minimum cardinality maximal matching?
I tried that pick a edge from highest degree vertex remove other edges from same vertex and so on.
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2answers
62 views
Either find a perfect matching, or return a witness that none exist [duplicate]
I am looking for a polynomial-time algorithm that takes as input a bipartite graph $(X\cup Y, E)$, and returns one of two options:
If a perfect matching exists, it returns the matching;
Otherwise, it ...
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2answers
48 views
Two Problems in understanding the algorithm for computing shortest paths in undirected graphs with possibly negative edge weights
Section 2 of this Lecture Note: Shortest Path Algorithms Luis Goddyn, Math 408 describes an algorithm using Edmonds' Minimum Weight Perfect Matching Algorithm to solve the shortest path problem for ...
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1answer
47 views
Minimum perfect matching with uneven vertices?
Given this graph, what is the minimum perfect matching? What do you do, when there is an uneven number of vertices?
2
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2answers
151 views
Christofides algorithm (by hand) (suboptimal solution - is it my fault?)
I would like to calculate an eularian path using Christofides algorithm on this graph: (Focus on the first number in each box representing the distance)
$\alpha$ denotes the start and end vertex of ...
0
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1answer
58 views
Stable matching with asymmetric arrays (gale shapley)
I was reading this thread The stable marriage algorithm with asymmetric arrays and started to solve the problem asked in this thread about matching 5 students with 10 dorms.
One of the answer ...
3
votes
2answers
247 views
Why does this greedy algorithm fail to accurately determine whether a graph is a perfect matching?
I came across this problem in Tim Roughgarden's course on Coursera:
In this problem you are given as input a graph $T=(V,E)$ that is a
tree (that is, $T$ is undirected, connected, and acyclic). A ...
3
votes
0answers
102 views
Edmond's Blossom algorithm (Maximum Matching) explanation
I asked this question on Math Stackexchange but it didn't get much attention, so I am asking it here.
Edmond's Blossom algorithm (Wikipedia), or simply the blossom algorithm, is a popular graph ...
3
votes
1answer
40 views
Maximizing quantities/length in buckets to match each other
I have a use case, real one, and I'm trying to come up with an efficient maximization algorithm to solve it.
I'll try to simplify, with a simple analogue, and after will explain the real world use ...
3
votes
1answer
27 views
Determine whether two collections of items can be paired
Given collections I (items) and S (slots), where I >= S.
And a pairing function that ...
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1answer
46 views
Obtaining Max-Weight Matching from Max-Weight-Max-Cardinality Matching?
I have a graph with integer-valued edge weights (possibly negative) on which I would like to obtain a maximum-weight matching.
However, I am using python-graph-tool, which only has max-cardinality ...
3
votes
1answer
649 views
Algorithm for matching people
I work at a summer camp where one of the activities is called Speed Dating. It's a game where participants talk with each other for a fixed amount of time. At the end everyone has to list three people ...
3
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1answer
226 views
Which algorithm can calculate an optimal allocation of students to projects?
I am trying to find an algorithm which calculates an optimal and stable allocation of $n$ students to $m$ projects, where each student strictly ranks all projects by preference. The available projects ...
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0answers
47 views
Hopcroft und Karp Algorithm and a suitable representation for a graph
Hey Guys I have a small question about the Hopcroft and Karp Algorithm.
I have a task to solve where I have to calculate the perfect matching. My professor told me I should use the Hopcroft Algorithm....
4
votes
1answer
100 views
Is a perfect matching's weight less than MST of a metric graph?
This is part of a bigger proof I'm trying to solve, which eventually came down to one thing:
Let $G=(V,E)$ an un-directed, complete, metric graph (maintains the triangle inequality) with an even ...
3
votes
1answer
96 views
Students in classroom problem - Flow in network
I have room, which is opened some days in week, in different hours each day.
I have multiple students, each has time some days in week, in different hours.
Each student have to visit the room ...
2
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2answers
35 views
Can a perfect matching always be found by a picking sequence?
There are $n$ people and $n$ items. For each person, there is a set of items he likes. Our goal is to give to each person a single item that he likes, i.e, find a perfect matching in the preference ...
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0answers
21 views
Is implementation of approximate dynamic maximal matching feasible?
I'm wondering if algorithms described in:
https://ieeexplore.ieee.org/abstract/document/7782946/
http://drops.dagstuhl.de/opus/volltexte/2014/4845/
easy to implement. They seem to mention the word "...
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1answer
38 views
Group tuples to satisfy constraints
This is a problem that involves matching students with various skills into groups so that there are as many groups as possible while ensuring that each group has certain skills present. I've reduced ...
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0answers
206 views
Best algorithm to generate matchmaking pairs
I am trying to come up with a matchmaking algorithm based on player ratings for my game, and I am pretty sure the algorithm I have is not the best (or maybe doesn't even work), but have no idea how to ...
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1answer
485 views
Which algorithm for game matchmaking in tournament
I organize amateur tournaments every week with random players.
Every week, players sign up, and I get a list with:
Name
Rank (1, 2 or 3)
Role (s) (DPS, TANK and HEALER)
I need to get balanced teams ...
2
votes
1answer
1k views
Perfect matching in a bipartite regular graph in linear time
Given a $G=(V,E)$ bipartite, undirected, 4-regular graph, I would like to find a perfect matching in linear time.
It is easy to show that there is a perfect matching for the graph, by using flow and ...
4
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1answer
571 views
Changing preference in Gale-Shapley algorithm?
Suppose, in the context of the classic marriage problem, two equal size groups of $n$ men and $n$ women are being matched, with the GS algorithm.
If a man were to switch the order of a pair of women, ...
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2answers
49 views
maximum matching number decreases when vertices collapse?
It seems that the maximum matching number decreases when some independent (not connected) vertices collapse into one vertex, but I don't know if it is absolutely true.
Would maximum matching number ...
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0answers
101 views
Complexity of removing edges to eliminate a perfect matching
Suppose $G$ is a bipartite graph which has a perfect matching. I want to find the fewest number of edges to delete from $G$ so that a perfect matching no longer exists. What is the complexity of this ...
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1answer
32 views
Decide if there is a sbugroup of edges in graph that every vertex meets exactly k edges from subgroup
I've been trying to solve the following question:
Show a polynomial algorithm for the following problem. Let $G = (V,
> E)$ a graph. The goal is to decide if there is $E' ⊆ E$ , such that
for ...
